Graphs of Trigonometric Functions

3 Key excerpts on "Graphs of Trigonometric Functions"

• 

  eBook - ePub

  Trigonometry For Dummies

  • Mary Jane Sterling(Author)
  • 2014(Publication Date)
  • For Dummies

    (Publisher)

  Part V

  The Graphs of Trig Functions

  Find out more about the polar coordinate system in an article at www.dummies.com/extras/trigonometry .

  In this part…

  • Create the basic graphs of the six trig functions.
  • Use the basic graphs of sine and cosine to more easily graph cosecant and secant.
  • Perform transformations on graphs of trig functions to make them fit a particular situation.
  • Use trig functions to model periodic applications — things occurring over and over as time goes by.

  Passage contains an image

  Chapter 19

  Graphing Sine and Cosine

  In This Chapter

  Looking at the basic graphs of sine and cosine

  Working with variations of the graphs

  Using sine and cosine curves to make predictions

  T he graphs of the sine and cosine functions are very similar. If you look at them without a coordinate axis for reference, you can't tell them apart. They keep repeating the same values over and over — and the values, or outputs, are the same for the two functions. These two graphs are the most recognizable and useful for modeling real-life situations. The sine and cosine curves can represent anything tied to seasons — the weather, shopping, hunting, and daylight. The equations and graphs of the curves are helpful in describing what happens during those seasons. You also find the curves used in predator-prey scenarios and physical cycles.

  The ABCs of Graphing

  You can graph trig functions in a snap — well, maybe not that fast — but you can do it quickly and efficiently with just a few pointers. If you set up the axes properly and have a general knowledge of the different functions’ shapes, then you're in business.

  Different kinds of values represent the two axes in trig graphs. The x- axis is in angle measures, and the y- axis is in plain old numbers. The x- axis is labeled in either degrees or radians. Often, a graph represents the values from –2π to 2π to accommodate two complete cycles of the sine, cosine, secant, or cosecant functions (or four complete cycles of the tangent or cotangent functions). If the x- axis is labeled in degrees, it typically ranges from –360 to 360, which is a wide number range. That range is in sharp contrast to the y- axis, which often just goes from –5 to 5. You'll find that radians — which are real numbers — are preferable when graphing trig functions. The y-

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  eBook - ePub

  Engineering Mathematics

  A Programmed Approach, 3th Edition

  • C W. Evans(Author)
  • 2019(Publication Date)
  • Routledge

    (Publisher)

  Fig. 3.4 ).

  The other circular functions, known as tangent, cotangent, secant and cosecant, can be defined in terms of cosine and sine. In fact

  tan     θ = 

  sin   θ

  cos   θ

          cot θ  =

  cos θ

  sin   θ

  sec   θ = 

  1

  cos   θ

            cosec   θ = 

  1

  sin   θ

  However, whereas cosine and sine have the real numbers ℝ as their domain, these subsidiary functions are not defined for all real numbers. □  Obtain the domain of each of these subsidiary circular functions by using the convention of the maximal domain.

  a  tan θ is defined whenever cos θ ≠ 0. From Fig. 3.4 we see that this is when θ is not an odd multiple of π/2. Any odd number can be written in the form 2n + 1 where n ∈ ℤ. Therefore the domain of the tangent function is

  A =

  {

  x |

  x     ∈     ℝ ,     x   ≠

  (

  2 n + 1

  )

  π / 2

  , n ∈ ℤ

  }

  b  cot θ is defined whenever sin θ ≠ 0. So θ must not be a multiple of π. Therefore the domain of the cotangent function is

  A =

  {

  x |

  x     ∈     ℝ ,     x   ≠ n π , n ∈ ℤ

  }

  Fig. 3.5 The tangent function.

  Fig. 3.6 The cosecant function.

  The domains of the secant and cosecant are the same as those of the tangent and cotangent respectively. ■

  The graph of y = tan x shows that the tangent function has period π (Fig. 3.5 ).

  The graph of the sine, cosine and tangent functions can be used to draw the graphs of the cosecant (Fig. 3.6 ), secant (Fig. 3.7 ) and cotangent functions. The graph of y = sec x has the same shape as the graph of y = cosec x. To obtain the graph of y = cosec x from the graph of y = sec x we merely need to relocate the y-axis through x = π/2 and relabel.

  Fig. 3.7 The secant function.

  You will remember that we write cos

  n

  θ instead of (cos θ)

  n

  when n is a natural number. This must not be confused with cos (nθ), and you should be alert to the fact that this notation does not hold good when n

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  eBook - ePub

  The Ellipse

  A Historical and Mathematical Journey

  • Arthur Mazer(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  y ) coordinates.

  Figure 5.2 Trigonometric functions as coordinates of a unit circle.

  The angle θ is sometimes expressed in radians and sometimes expressed in degrees. It is worth one’s effort to be able to use both units of measurement.

  5.1.2 Triangles

  For angles between 0˚ and 90˚ (from 0 to π /2radians), the trigonometric functions correspond to ratios of right triangles as illustrated in Figure 5.3 .

  Figure 5.3 Trigonometric functions as triangular ratios.

  The equivalence between the definitions in the preceding two tables is seen by noting the triangle formed between a point on the unit circle, the origin, and the point along the x axis given by the x coordinate of the original point. The ratios that define the trigonometric functions are the same for all similar triangles.

  5.1.3 Examples

  Using the definitions, it is possible to determine the trigonometric functions for some values of θ . Examples are given below. These examples are the first entries into a trigonometric table that is further developed in subsequent sections.

  Example 5.1

  Determine the trigonometric for the value θ = π /4 rad (45˚).

  Solution

  When the angle θ is π/4 rad, x = y along the unit circle (see Figure 5.4 ). With the assistance of the Pythagorean theorem, the values for x and y

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